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3 votes

No, if $P $ is False, then $P → Q$ is always $True.$

Lets take an example to understand: 

“If India win wins the world cup, I will give party”

Here the condition is “If India wins” (P) then only “I will give party” (Q).

But lets say $P$ is $False$ means India failed to win the World cup, then I give the party or not, it doesn’t impact on the condition of the statement. Its still True i.e $P → Q$ is  $True.$ 

 

(** Condition would be false when India wins the world cup, but still I don’t give party )

1 votes
1 votes

P → Q
 

If anything is given as  r: $p→q$  then the propositional formula ‘r’ or the compound proposition may or may not be true.
Now u r given that P is False 
Lets break down implication : $p→ q$ means $p’ + q$  as both of their truth tables are exactly same
Now we can go for the truth table only with p as F 

P q
                             F                                T
                             F                                F

 


 

Now compute $p’ +q$ we will get T always so p→ q is always true if p is true.

NOTE : p-->q is always true iff p is false or q is true

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Here, it doesn’t really matter what the implicated statement (here, q) is, if the premise (here, p) itself is false. The implication has not been proven false. The whole thing p→q is still true.

You’ll understand it better with an example from Rosen’s book :

A useful way to understand the truth value of a conditional statement is to think of an obligation or a contract. For example, the pledge many politicians make when running for office is
“If I am elected, then I will lower taxes.”

If the politician is elected, voters would expect this politician to lower taxes. Furthermore, if the politician is not elected, then voters will not have any expectation that this person will lower taxes, although the person may have sufficient influence to cause those in power to lower taxes. It is only when the politician is elected but does not lower taxes that voters can say that the politician has broken the campaign pledge. This last scenario corresponds to the case when p is true but q is false in p → q.

Similarly, consider a statement that a professor might make: “If you get 100% on the final, then you will get an A.”
If you manage to get a 100% on the final, then you would expect to receive an A. If you do not get 100% you may or may not receive an A depending on other factors. However, if you do get
100%, but the professor does not give you an A, you will feel cheated.

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